The previous article described equal temperaments as mathematical
series. This article will describe them as two-dimensional mathematical matrices.

Exact musical octaves, 2/1 ratios, are among the results of the formula

Fm = 2m/kFR

in which m and k are integers. The octaves are as

2nFm = Fm+nk/Fm =
2 (m + nk)/kFR

and we may also revise the formula to

2nFm = 2(m/k) + nFR

Using this revised formula, we may generate the octaves through a separate
variable, n. To avoid generating redundant solutions, we also require that mH
- mL < k.

If mH = k - 1 and mL
= 0, the k values of m define the equal-tempered pitches within
one octave. By using different integer values of n, we may transpose these pitches
to other octaves and define an equal-tempered scale of as many octaves as we wish.
Mathematically speaking, we are now defining the equal temperament as a two-dimensional
matrix, even though it can be defined more simply as a one-dimensional series according to
our earlier formula. The more complicated formula is useful because it is often desirable
in music to categorize musical pitches in groups of octaves.

The table below gives the nominal fundamental frequencies of the notes sounded
by the 88 keys of a piano. The horizontal dimension represents octave intervals, and the
vertical dimension represents semitones within each octave. The mathematical derivation of
the octaves is shown in red, and that for the semitones within each octave, in green. The
reference frequency, A, 440 Hz, is highlighted in a yellow table cell.

m
=

2m/12

Fm,n

cents

note name

11

211/12

51.91

103.83

207.65

415.30

830.61

1661.22

3322.44

1100

G#/Ab

10

25/6

49.00

98.00

196.00

392.00

783.99

1567.98

3135.96

1000

G

9

23/4

46.25

92.50

185.00

369.99

739.99

1479.98

2959.96

900

F#/Gb

8

22/3

43.65

87.31

174.61

349.23

698.46

1396.91

2793.83

800

F

7

27/12

41.20

82.41

164.81

329.63

659.26

1318.51

2637.02

700

E

6

21/2

38.89

77.78

155.56

311.13

622.25

1244.51

2489.02

600

D#/Eb

5

25/12

36.71

73.42

146.83

293.66

587.33

1174.66

2349.32

500

D

4

21/3

34.65

69.30

138.59

277.18

554.37

1108.73

2217.46

400

C#/Db

3

21/4

32.70

65.41

130.81

261.63

523.25

1046.50

2093.00

4186.01

300

C

2

21/6

30.87

61.74

123.47

246.94

493.88

987.77

1975.53

3951.07

200

B

1

21/12

29.14

58.27

116.54

233.08

466.16

932.33

1864.66

3729.31

100

A#/Bb

0

20

27.50

55.00

110.00

220.00

440.00

880.00

1760.00

3520.00

0

A

2n=

1/16

1/8

1/4

1/2

1

2

4

8

2n

2-4

2-3

2-2

2-1

20

21

22

23

n=

-4

-3

-2

-1

0

1

2

3

In an actual piano, the partials of the strings are inharmonic, due largely to bending
stiffness, and so the scale is slightly stretched to minimize beating. In the table below,
for example, the lowest A (cell with blue type) is 26.95 Hz, rather than the 27.5 Hz which
is exactly 1/16 of the reference A, 440 Hz.

The scale of the piano is stretched more at
its extremes, where the thickness of the strings is greater in relation to their length,
increasing the bending stiffness. The formula applied in the table below modifies the
nominal, equal-tempered fundamental frequencies as:

Fs = (F(m,n) /440)1.002 · 0.1[log2
(F(m,n) /440)3]

where Fs is the stretched frequency and F(m,n) is
the nominal frequency. With a nominal equal-tempered scale where FR = 440, the
above formula is equivalent to:

Fs = (F(m,n) /440)1.002 · 0.1[(m
+ n/12)/440]3]

This formula is intended only as an example, since the actual stretching depends on the
characteristics of the strings of each individual piano. (Another
article on this site describes the vibrations of stretched wires in more detail. The
topic of that article is the tensioning of bicycle spokes, which, as vibrating systems,
are essentially identical with piano strings.)